Direct controllability and observability time domain ... - IEEE Xplore

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 33, NO. 8, AUGUST 1988

Using (2.12). (2.17), and partitioning So as (2.18)

where dim SI = n x n, dim S2 = n x r, and dim S3 = r x r , from (2.16) we get

1

matrix via a Leverrier-type algorithm. Hence, no decomposition of the state-space model is needed and the criteria are easily implementable on a digital computer.

I. INTRODUCTION Consider the linear, time-invariant system of the form

I-

.Y‘

1

EX(t) = A x ( [ )+ Bu(t)

(A+BJ)Si+Sj(A+BJ)’+BHS;+S,H‘B’+-DD‘ Y

for a l l y E R” and fi > 0. Now, as SI is a positive-definite matrix, so is P : = S;’. defining X:= J+HS;P

Therefore, (2.20)

and making the transformation f : = Sly, from (2.19) and (2.20) it follows that:

Y

(14

where E is an n x n singular square matrix, x ( t ) is an n-dimensional vector of states, u ( t ) is an m-dimensional vector of inputs, y ( t ) is a pdimensional vector of outputs, and A , B, and C are matrices of appropriate dimensions. The system of the form (1) is called singular or semistate, or generalized state-space or descriptor. For an extensive literature on singular systems, see [I]. The controllability of singular systems has been studied extensively in recent literature. In [2], [3] controllability conditions for singular systems are established in the frequency domain, where the matrix

Y

for all f E Eln and some positive constant 6. Therefore, using Proposition 2.6 and Theorem 2.5, there exists a controller of form (1.2) [given by (2.2011 such that the system C is stabilizable with disturbance attenuation y. Since ys in (1.4) is the infimal norm via static state-feedback, it follows that ys 5 y. Now, recalling the definition of y from (2.13), we obtain that yr 5 Y d 26. Finally, since E is arbitrary, it follows that ys 5 yd, which 0 proves the desired result. Remark 2.22: We would like to note that Theorem 2.1 in conjunction with the results of Khargonekar and Poolla [3] can be used to prove the following fact: the infimum of the H , norm of the closed-loop transfer function using internally stabilizing static state-feedback is no greater than the infimum of the operator norm of the closed-loop transfer function using internally stabilizing nonlinear time-varying state-feedback.

+

is checked for full rank. In the time domain, [4]-[7] introduce criteria for controllability and observability conditions needed in the decomposition of the system to the Weierstass form, whereas [8]gives the criteria using the well-known geometric techniques. In this note we present new direct controllability and observability criteria for singular systems, using a direct expansion of the transfer function matrix. The criteria are computationally more efficient than existing ones in the time domain, because there is no need for decoupling the system.

11. PRELIMINARY RESULTS The transfer function matrix of system (1) can be written in the form [9]

REFERENCES

H(s) = C[SE- A ] ’ B

[ l ] B. A. Francis, A Course in H , Control Theory. New York: Springer-Verlag. 1987. [2] D. H. Jacobson, Extensions of Linear-Quadratic Control, Optimization and Matrix Theory. New York: Academic, p. 1977. [3] P. P. Khargonekar and K. R. Poola, “Uniformly optimal control of linear timeinvariant plants: Nonlinear time-varying controllers,” Syst. Confr. Lett., vol. 6, no. 5. pp. 303-308, Jan. 1986. 141 I. R. Petersen, “Disturbance attenuation and H , optimization: A design method based on the algebraic Riccati equation,” IEEE Trans. Automot. Confr., vol. AC-32, no. 5, pp. 427-429, May 1987.

Direct Controllability and Observability Time Domain Conditions of Singular Systems B. G. MERTZIOS, M. A. CHRISTODOULOU, B. L. SYRMOS, AND F. L. LEWIS

(3)

where R ( s ) = adj [SE - A ] and q ( s ) is the characteristic polynomial of the given system which is written in the form

The constant coefficient matrices R, ,,k , k = 0, 1, . . . , n - 1 and the constant coefficients qnk,k = 0, 1 , . . ., n are calculated by the following recursive algorithm [9]: ~

A bstract-New controllability and observability criteria for singular systems using direct formulas in the time domain are presented. The method used here is based on a direct expansion of the transfer function Manuscript received November 6, 1986; revised November 29, 1986. June 17, 1987, and October 19, 1987. B. G . Mertzios is with the Department of Electrical Engineering, Democritus University of Thrace, 67100 Xanthi, Greece. B . L. Syrmos and F. L. Lewis are with the School of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332. M. A. Christodoulou is with the Department of Electrical and Computer Engineering, Syracuse University, Syracuse, NY 13210. IEEE Log Number 8821567.

with boundary conditions

0018-9286/88~0800-0788$01 .OO

Rm = I , R_,,,=R,,_,=O, &=O,

0 1988 IEEE

for i > l , k r l

for k > i r 0 .

(7b) (74

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 33, NO. 8, AUGUST 1988

Then

111. CONTROLLABILITY

It can be seen from (3) that the transfer function between the state and the input is given by

e may be written in the form

( ? = [ [ ( A " - 'E, ' ) B + ~ I O ( A "Ea)B+ - ~ , . . . +qn-t.o(A0,Eo)BII

1

G (s) =-R (s)B 4 (4

Now we can state the following theorem. Theorem I : The singular system (1) is completely controllable if and only if the n x nm matrix (? = [R,-

~

R,- ] , ] B ~

. ..

~

R,-

(9)

,BI

From (1 1) it is seen that C? may be written as a product of two matrices

e=s.p

(12)

where S and P are given by

is of full rank. Proof: It is based on the fact that the equation

can be solved for U@) iff

e is of full rank.

( - l ) n - l ( A oE, n - ' ) B ]

P=

The matrix C? can be computed using the algorithm ( 5 ) , (6). However, the matrix C? can be written in a more convenient form, by writing the matrices R,,I , k in (4) in the form [IO] R,_ I J =( - 1)' [ ( A'-'-I,

E') + qIo(A"-'-*, E')

+ . . . + qn-,-

1.0

',

( A E')]

+ ( - l ) ' - l [ q l l ( A n - ' - lE , J - I )+q21(An-'-2, EJ-I) + . . . + q n - , ,](AO,E ' - ] ) ]

+ [q,,(A"-'-l,

+

EO) q,+ ','(A"-'-Z, EO) + . . . + q n - l , , ( , 4 0 , EO)]

Equation (10) is used to establish the following theorem. Theorem 2: The system (1) can be completely controllable only if matrix S is of full rank. Proof: Since the matrix P i s always of full rank (note that the lowest block of P is an nm x nm unity matrix), the matrix e can be of full rank only if S is of full rank. Note that the matrix P has more rows than columns. Thus, the full rank of S and P does not ensure that C? is also of full rank. Therefore, the condition given in Theorem 2 is only necessary but not sufficient. W . OBSERVABILITY

The transfer function between the output and the state is given by k=O

1=0

where the term ( Ak , E') denotes the summation of all possible products of matrices A and E where A appears exactly k and E I times in each product [ll], as follows: ( A k ,E ' ) = A k E 1 + A k - I E ' A +' . . E I A k .

Using analogous reasonings as in Section III, we establish the following theorems.

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 33, NO. 8, AUGUST 1988

Theorem 3: The singular system (1) is completely observable if and only if the p m x n matrix

The observability matrix 0 is given by

r

is of full rank.

-1

3

-5

0

1

1

Lo

0

0

o=

Theorem 4: The system (2) can be completely observable only if the matrix S is of full rank where 0 = P . s.

The matrix S is given by which is of full rank. Thus, the system is completely observable. The matrices S and S are given by

s

[

=

1

0

o 1

1

1 1

2

1 0

0

-1

-

1

0

2

-1

0 -1

and P is exactly the transpose of P, i.e., P = PT. V. ILLUSTRATIVE EXAMPLE

L 1

-3

-1

Let a system of the form (1) be given by which are also of full rank. VI. CONCLUSIONS In this note, time-domain criteria for controllability and observability of singular systems were presented. Necessary and sufficient, as well as only necessary conditions are developed without decomposing the generalized state-space model via the Weierstass, Drazin, or singular value decomposition forms [12]. Thus, the proposed time-domain conditions are computationally more efficient than the existing ones, which are discussed in [3]. Moreover, the compact form of the criteria matrices may be helpful in other analysis as well as synthesis problems in singular systems.

L’l Applying the presented algorithm, we obtain

[ -:] [ -:1; -U]

1 0 RZO= - 11 0I

-1

, R2,=

0

,

-1

REFERENCES

and

The matrix

e is given by e=

[

0

-2

1 I

1

3 1 - 1

: I

- 1 ; o

- 1 ; - 1 I

-1

;

0

0

1

- 2 : o

0

-1

which is of full rank. Thus, the system is completely controllable.

F. L. Lewis, “A survey of linear singular systems,’’ Circuits, Syst., Signal Processing (Special Issue on Semistate Systems), vol. 5 , no. 1, pp. 3-36, 1986. H. H. Rosenbrock, “Structural properties of linear dynamic systems,’’ Int. J. Contr., vol. 20, no. 2, pp. 191-202, 1974. G. C. Verghese, B. C. Levy, and T. Kailath, “A generalized state space for singular systems,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 811-831, 1981. E. L. Yip and R. F. Sincovec, “Solvability, controllability and observability of continuous descriptor systems,” IEEE Trans. Automat. Contr., vol. AC-26, no. 3, pp. 702-707, 1981. L. Pandolfi, “Controllability and stabilization for linear systems of algebraic and differential equations,” J. Optimiz. Theory Appl., vol. 30, pp. 601-620, 1980. M. A. Christodoulou, “Analysis and synthesis of singular systems,” Ph.D. dissertation, Dep. Elect. Eng., Democritus U. of Thrace, Xanthi, Greece, 1984. D. Cobb, “Controllability, observability and duality in singular systems,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 1076-1082, Dec. 1984. K. Ozcaldiran, “A geometric characterization of the reachable and the controllable subspaces of descriptor systems,’’ Circuits Syst., Signal Processing (Special Issue on Semistate Systems), vol. 5 , no. 1, pp. 3-36, 1986. B. G. Mertzios, “Leverrier’s algorithm for singular systems,’’ IEEE Trans. Automat. Contr., vol. AC-29, pp. 652-653, July 1984. B. G . Mertzios and B. L. Syrmos, “Transfer function matrix of singular systems,” IEEE Trans. Automat. Contr., vol. AC-32, pp. 829-831, Sept. 1987.

79 1

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 33, NO. 8, AUGUST 1988 [ l l ] B. G. Mertzios and M. A. Christodoulou, “On the generalized Cayley-Hamilton theorem,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 156-157, Feb. 1986. [12] F. R. Gantmacher, The Theory of Matrices. New York: Chelsea, 1959.

i.e.,

H ( s )=

1

l+qs+-

[e

1

012.9

+.

1 ff2s+ffG+ .

+i ff“S Scaled Impulse Energy Approximation for Model Reduction T.N. LUCAS Abstract-It is shown that the impulse energy approximation technique for model reduction may be improved by scaling in the frequencydomain. The method remains simple to use and retains the stability property of impulse energy approximation. Criteria are given for choosing an appropriate scaling parameter and an example illustrates the technique.

I. INTRODUCTION Recently, interest has been generated in using impulse energy approximation for linear system reduction [1]-[3]. The technique is a simple variation of the Routh method [4] and amounts to retaining the k pairs of alpha and beta parameters which contribute most to the reduced kth-order model’s impulse response energy. Lucas [l] shows that sometimes it is possible to obtain good approximations using this approach when the Routh method fails (because of the energy distribution) without resorting to more computationally sophisticated methods. However, Rao [2] expressed concern over how to choose the most significant energy parameters in cases where equal or nearly equal values of e, = a,occur. Lucas [3] suggested that this problem might be overcome by attaching more importance to parameters associated with the lowfrequency behavior of the system, for step response approximations, giving rise to the idea of “weighted” energy approximants. The purpose of this correspondence is to introduce the idea of scaled impulse energy approximation for model reduction. It is shown that the method is a logical extension of previous work [l], [3] and makes the approach of impulse energy approximation more flexible, thus widening its appeal. The technique is seen to retain the desirable properties of stability preservation and computational simplicity while in addition, by appropriate choice of a scaling parameter, it is possible to retain the full system impulse energy in the reduced model or match the initial Markov parameter or even find an optimal scaled reduced model. A numerical example is given to illustrate the application of the method.

[$I

2 + . . . +

. .]I

+i a“S

a,’+m,’andPl’+rP,’

( i = l , 2,

..., k )

Consider the nth-order linear system, in usual notation, given by the transfer function G ( s )=

bo+ bls+

. . . + b,- IS“-’ . . . + a,s“

Hk(s)

rH&)

(4)

+

and consequently, (5)

Thus, if G&) is given in standard notation by

then ( 5 ) and (6) show that the scaling transformation (3) implies

c,+c,/ri

Reduction of G(s) to kth-order by impulse energy approximation [l] requires calculating the parameter pairs (a,, 0,) i = 1 , 2 , . . .,n, in the a0 expansion [4] of the reciprocate transfer function H(s) = (l/s)G(l/s), Manuscript received September 9, 1987; revised November 17, 1987. The author is with the Department of Mathematics and Statistics, Paisley College of Technology, Paisley, Scotland. IEEE Log Number 8821569.

i = O , 1, 2,

d,+di/ri

..., k-1

i = O , 1, 2,

..., k

(7)

giving the coefficients of the scaled approximation Gk(s/r). Before discussing any advantages of the scaled energy approximant it is interesting to notice that i) in the frequency domain, the scaling effectively multiplies the magnitudes of the poles and zeros of the impulse energy approximant by r, and ii) the effect of this in the time domain can be seen from the theory of Laplace transforms using the well-known result

G-’{Gk

ao+als+

(3)

where (a,’, 0,’) are the retained parameter pairs from the full system. The first thing to notice is that the a-parameters of the reduced model will remain positive, and hence stability of the model is guaranteed for arbitrary r > 0. Also, if Gk(s)is the transfer function of the k-th order impulse energy approximant [l], obtained from (a,’, 0,’) i = 1 , 2, * .., k, and H k ( s ) = (l/s)G(I/s) is its reciprocate, then it can be seen from the a-8 expansion of H k ( s ) [similar to that in (2)] that the scaling transformation (3) implies

of/

11. SCALED IMPULSE ENERGY APPROXIMATION

(2)

which is conveniently accomplished by the so-called a- and 0-tables [4], or the D- and N-tables [I]. Values of the energy parameters e, = ofla,( i = 1 , 2, 3 , .. . , n) are then compared and the k largest values are retained with their corresponding (a,0) parameters to form the reduced model. For ensuring a correct steady-state response for a step input, a slight PI) must be retained along with modification might be needed in that (a,, the other (k-1) most significant parameter pairs [l], [3]. Consider now what happens if each of the retained a and 0 parameters in the reduced model is multiplied by a scaling parameter r > 0, i.e.,

(

f))=rgx(rt),

wheregk(t)=C-l{Gx(s)},

i.e., time-scaling occurs.

In. CHOICE OF SCALING PARAMETER Since the choice of the scaling parameter r is quite arbitrary for stability preservation, it might be desirable to have some criterion to help choose a suitable value. Here, three such criteria are suggested and serve to demonstrate the flexibility of the method.

oO18-9286/88/0800-0791$Ol.oO 0 1988 IEEE